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A classification of the solutions of a differential equation according to their asymptotic behaviour

Published online by Cambridge University Press:  14 November 2011

Uri Elias
Uri Elias
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, U.S.A.
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Synopsis

The solutions of the differential equation Lny + p(x)y = 0, where Lny = ρn(ρn−1 … (ρ1(ρ0y)′)′ …)′ and p(x) is of one sign, are classified according to their behaviour as x → ∞. The solution space is decomposed into disjoint, non-empty sets Sk, 0≦K≦n, such that (−1)n−kp(x)≦0. We study the growth properties and the density of the zeros of the solutions which belong to the different sets Sk, the structure of the sets and its connection with (k, n − k)-disfocality.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 25 - 38
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Ahmad, Shair and Lazer, A. C.. On the oscillatory behavior of a class of linear third order differential equations. J. Math. Anal. Appl. 28 (1970), 681–689.CrossRefGoogle Scholar
2Dolan, J. M. and Klaasen, G. A.. Dominance of n−th order linear equations. Rocky Mountain J. Math. 5 (1975), 263–270.CrossRefGoogle Scholar
3Elias, U.. Oscillatory solutions and extremal points for a linear differential equation. Arch. Rational Mech. Anal, (to appear).Google Scholar
4Etgen, G. J. and Taylor, W. E.. The essential uniqueness of bounded nonoscillatory solutions of certain even order differential equations. Pacific J. Math. 68 (1977), 339–346.CrossRefGoogle Scholar
5Hanan, M.. Oscillation criteria for third order linear differential equations. Pacific J. Math. 11 (1961), 919–944.CrossRefGoogle Scholar
6Hartman, P. and Wintner, A.. Linear differential and difference equations with monotone solutions. Amer. J. Math. 75 (1953), 731–743.CrossRefGoogle Scholar
7Jones, G. D.. Oscillatory behavior of third order differential equations. Proc. Amer. Math. Soc. 43 (1974), 133–136.CrossRefGoogle Scholar
8Jones, G. D. and Rankin, S. M.. Oscillation properties of certain self-adjoint differential equations of the fourth order. Pacific J. Math. 63 (1976), 179–184.CrossRefGoogle Scholar
9Kiguradze, I. T.. Oscillation properties of solutions of certain ordinary differential equations. Soviet Math. Dokl. 3 (1962), 649–652.Google Scholar
10Kim, W. J.. Nonoscillatory solutions of a class of n−th order linear differential equations. J. Differential Equations 27 (1978), 19–27.CrossRefGoogle Scholar
11Kondrat'ev, V. A.. Oscillatory properties of solutions of the equation y′″ + p(x)y = 0. Trudy Moskov. Mat. Obšč. 10 (1961), 419–436.Google Scholar
12Lazer, A. C.. The behaviour of solutions of the differential equation y′″ + py′ + qy = 0. Pacific J. Math. 17 (1966), 435–466.CrossRefGoogle Scholar
13Staikos, V. A. and Sficas, Y. G.. Oscillatory and asymptotic characterization of solutions of differential equation with deviating argument. J. London Math. Soc. 10 (1975), 39–47.CrossRefGoogle Scholar
14Švec, M.. Sur une propriété des intégrales de l'équation y(n) + Q(x)y = 0, n = 3,4. Czechoslovak Math. J. 7 (1957), 450–462.CrossRefGoogle Scholar
15Trench, W. F.. Canonical forms and principal systems for general disconjugate equations. Trans. Amer. Math. Soc. 189 (1974), 319–327.CrossRefGoogle Scholar